Factors of 31 | Prime Factorization of 31 - Explained Simply

Today we are going to present here Factor Tree of 31. The factor is the number that divides the original number. The Factors of 31 are 1 and 31 itself.

What are the factors of 31?

To find the factors of 31, we look for all the whole numbers that can divide 31 evenly without leaving a remainder.

Factor Tree Method of 31: Explained Simply

Now, let's put our factor-finding strategy into action for the number 31. We'll use the trial division method.

Start with 1:

 31 ÷ 1 = 31

Factors found: 1 and 31

Check 2:

 31 ÷ 2 = 15.5

Since 15.5 is not a whole number, 2 is not a factor of 31.

Check 3:

 31 ÷ 3 = 10.33...

Not a whole number, so 3 is not a factor of 31.

Check 4:

 31 ÷ 4 = 7.75

Not a whole number, so 4 is not a factor of 31.

Check 5:

 31 ÷ 5 = 6.2

Not a whole number, so 5 is not a factor of 31.

Why can we stop here?

The square root of 31 is approximately 5.56. According to our tip, we only need to check numbers up to this point. Since we've already checked 5 and found no new factor pairs, and the next whole number is 6 (which is greater than 5.56), we can confidently stop. If 31 had another factor, we would have found it as a pair with one of the numbers we already checked (1, 2, 3, 4, or 5).

Therefore, the only whole numbers that divide 31 evenly are 1 and 31.

Summary of Factors for 31:

The fact that 31 only has two factors (1 and itself) points to a very important mathematical concept: Prime Numbers.

Prime Factorization of 31

Now let's apply our knowledge to the number 31.

To find the prime factorization of 31, we need to check if it has any factors other than 1 and itself. We can try dividing 31 by small prime numbers, starting from the smallest:

Is 31 divisible by 2? No, because 31 is an odd number.

Is 31 divisible by 3?

No, because 3 + 1 = 4, and 4 is not divisible by 3.

Is 31 divisible by 5?

No, because 31 does not end in a 0 or a 5.

Is 31 divisible by 7?

No, 7 × 4 = 28 and 7 × 5 = 35.

Is 31 divisible by 11?

No.

Key Insight for Checking Primes: When looking for prime factors, you only need to test prime numbers up to the square root of the number you are factoring.

The square root of 31 is \(\sqrt{31} \approx 5.57\).

This means we only need to check prime numbers less than or equal to 5.57. These primes are 2, 3, and 5.

Since we've already checked 2, 3, and 5 and found that none of them divide 31 evenly, we can confidently conclude that 31 has no prime factors other than 1 and itself.

Because 31 has only two factors (1 and 31), by definition, 31 is a prime number.

Therefore, its prime factorization is simply 31.